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### Trigonometry and Music

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Date: 12/21/98 at 09:57:17
From: Elizabeth
Subject: Trigonometry and music

How is trigonometry used in music?
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Date: 12/21/98 at 13:04:53
From: Doctor Santu
Subject: Re: Trigonometry and music

Dear Elizabeth,

Certainly the functions sine and cosine have a connection to music.

You probably know that sound consists of waves, and that sin(x) and
cos(x) are waves. Fourier, a French heat engineer, wondered whether it
might be possible to write *any* wave in terms of combinations of sines
and cosines; and it turned out that, yes, it's possible (provided your
wave is smooth and well-behaved enough).

So, for instance, a wave might be representable as a combination such
as  1.4 cos x + 2.8 sin x - 1.103 cos 2x + 3.32 cos 3x + 5.6 sin 3x.

Normally, a wave selected at random can only be represented by an
infinite number of such terms; in other words, the combination doesn't
stop, but goes on forever, though the numbers in front of the sines
and cosines get smaller and smaller.

Musically, when an instrument plays a note, the basic note can be
represented by A.cos Lt + B.sin Lt. The number L has to do with how
high the note is (pitch), and A and B have to do with how loud it is;
t is time.

But that function is a "pure tone" that sort of sounds like a flute.
The unique sounds of oboes and clarinets and so forth are due to almost
inaudible notes at the octave, octave+fifth, etc.

The Dr. Math archives contain a reply sent to a similar question
earlier this year:

http://mathforum.org/dr.math/problems/angelica12.3.97.html

Hope it's useful!

- Doctor Santu, The Math Forum
http://mathforum.org/dr.math/
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Date: 06/30/2002 at 21:54:29
From: Dan
Subject: Trigonometry and Music

I am aware that sound waves can be described through trigonometric
functions, in equation form. I am interested in how their geometric
shapes relate to their trigonometric functions. Why are certain
combinations of tones more pleasing to the ear than others? Can the
trigonometric function for a sound wave be used to describe light,
radio, X-ray waves?  How can we derive the function for a wave?
(example--sound wave)

Brian
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Date: 07/16/2002 at 09:10:44
From: Doctor Nitrogen
Subject: Re: Trigonometry and Music

Hello, Brian:

If you look at the graph for the sine and cosine functions, you will
see that they have all the characteristics of a wave traveling along
the x-axis. Just like a wave, they have amplitude, and they are
periodic, and even have a frequency.

There is a branch of Mathematics called Fourier Analysis/Fourier
Series, which uses this property of the sine and cosine graphs, to
decribe wave motion. Frequently Fourier Series are used to analyze the
wave properties of a vibrating string, like a violin string, or the
vibrational modes of a drum membrane. When Fourier Series are used
this way, they are usually solutions to Partial Differential
Equations.

Sound waves are longitudinal, rather than transverse, like water
waves, or waves along a string fixed at both ends and set to vibrate,
but sine and cosine waves describe sound waves also.

The reason certain tones are more pleasing to the ear than others is
that when partial differential equations describing wave motion are
solved by Fourier Series, the solutions have a special number
associated with them for the modes of vibration for the system. When
the vibrational modes are integral multiples of a fundamental mode of
vibration in the solution, you get music. When they are not, you
don't. That's why you can get harmonics on a violin, flute, a pipe
organ, or a piano, but not on a snare or bass drum. So classical
violinist Itzhak Perlman might be creating music with his violin, but
the rock star Bon Jovi's drummer is not.

Light waves were first described by a physicist named James Clerk
Maxwell. When he solved one of his partial differential equations he
got a wave motion for light. A branch of physics called Quantum
Mechanics, however, was developed around 1900 to explain certain
things about light waves that Maxwell's equations could not. Can you
believe it was once thought that because light was a wave there should
be no limit to the frequency of the wave in a physical system? This
meant that for ordinary light trapped in a box lined with mirrors, the
light could eventually escape as gamma rays! This paradox was called
Quantum Mechanics has replaced the old ideas about the wave nature of
light and replaced them with the idea that sometimes light behaves
like a particle, and sometimes like a wave.

In Quantum Mechanics, applied to the hydrogen atom, when an electron
"jumps" from a higher energy level to a lower one, the light energy it
radiates as a "photon" particle is:

E_2 - E_1 = n*hbar*nu, n = 1, 2, 3, ....

where hbar is a constant called h, divided by 2*pi, and nu is the
frequency of the photon.

Even in Quantum Mechanics there is a wave equation, called the
Schroedinger Wave Equation.

- Doctor Nitrogen, The Math Forum

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